First we observe that the kinetic pseudoenergy has scale dimensions
which is precisely opposite to that of the
potential term,
. This is the same dimensional situation as in the
harmonic oscillator,
where the dimensions are 
and 
, respectively. We
therefore decide to
go over to ``square root coordinates'' via some transformation
which
changes 
[11].
In two dimensions,
the appropriate square root is given by the Levi-Cività transformation
We may imagine the vectors 
to move in the complex
planes 
. Then the new variables
u corresponds to
the complex square root,
Let us also introduce the matrix
and write (74) as a matrix equation
In contrast to the three-dimensional
case to be treated below we
note here the rather obvious fact that the Levi-Cività mapping,
which is simply a transformation to parabolic coordinates,
carries the flat 
space into a
flat 
space. Indeed, from (74) we have the infinitesimal
transformation
with the basis dyad
and the reciprocal dyad
We therefore find for the connection,
the matrix elements 
as follows
Notice that the connection satisfies an important identity
which is seen to be a consequence of the defining relation
together with the obvious identity
and the diagonal property of
.
It can be shown quie generally that this is
the essential reason
for the absence of the time slicing corrections
to be proved in this section.
Both, torsion and
Cartan curvature tensor vanish identically, the latter
due to the
linearity of the basis dyads
in 
, which guarantees trivially
the integrability conditions
and thus 
In the continuum limit it is easy to see that the
Levy-Cività transformation
converts the action (73) into that of a
harmonic oscillator. With
we find
Apart from the trivial term 
,
this is the action of a harmonic oscillator,
which oscillates in the pseudotime s with an effective mass
and a pseudofrequency
Note that this 
has the
same dimension as 
pseudotime s corresponding to 
(in contrast to usual frequencies with 
).
The path integral is well defined only as long as the energy E of the H atom is negative, i.e., in the bound-state regime. The amplitude in the continuum regime with positive E will be obtained by analytic continuation.
In the regularized form we can now
calculate the pseudotime sliced amplitude.
Taking for the splitting parameter 
and ignoring, for a
moment, all complications
due to the finite time slicing, we deduce from (66),
and hence
Writing the integrals over the
's in (71) as
integrals over 
's we
see that the transformed amplitude (71) becomes simply
where 
denotes the time sliced
oscillator amplitude
The integrals can all be done with the known result, in the limit 
,
The symmetrization in 
in (94)
is necessary since for each path
from 
to 
there are two paths in the square-root
space, one from 
to 
and one from 
to 
.
The fixed-energy amplitudes are obtained by integrating the pseudotime displacement amplitude over all S.
Inserting (96), this becomes
with the abbreviation
which is the one-dimensional fluctuation factor
[see (2.145)]. The 
's on the right-hand side are related to the
's on the left by
In doing the integral we have to circumnavigate the singularities
in 
in accordance with the 
-prescription 
. This can be avoided by rotating the contour of S-integration
so that it runs along the negative imaginary semi-axis,
This amounts to going over to the euclidean amplitude of the harmonic oscillator. The amplitude can be rewritten in a pleasant form by introducing further the variables
Then
and we obtain
This can be used to find the energy spectrum and the wave functions.
Notice that the integral converges only for 
.
It is possible to write down another integral representation
which converges for all 
.
For this we change the variables of integration
to
so that
This gives
The integrand has a cut in the complex 
-plane from z=-1 to 
and from
to 
, with the integral running along
the right-hand cut. This is transformed
into an integral along a contour C
which encircles the right-hand cut in the clockwise sense.
Since the cut is of the
type 
, we have the replacement rule
and obtain
